# Full text of "Treatise On Analysis Vol-Ii"

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```430       XV    NORMED ALGEBRAS AND SPECTRAL THEORY

for all y e dom(H), which shows that H 4- af/is injective and that the inverse
mapping of Fa onto dom(H) is continuous (5.5.1). Also, since the operator
(H + a/7)"1 is continuous and its graph is closed (15.12.5), Fa is a closed sub-
space of E (15.12.2). If in (15.13.4.4) we put a = - 1 and y = (H + U)'1 • x,
where x e Fl9 then we get \\(H - //) .* y\\2 = \\(H + U) • j||2, hence || V • *|| =
M for all x e Fr, Since the subspace V(Fi) is therefore isometric to Fx, it is
complete and therefore c/oseJ (3.14.4). On the other hand, the relations
H • y + iy = x and H - y — iy = K • x give

/y = K*- F'*)»        #'J> = i(* + V-x)

for all jc e Ft. This shows that, if x — V • x = 0 for x e Fl9 then we must have
jt + y- x = Q, hence * = 0. Hence (i) and (ii) are proved.

We shall now prove (iii). The hypothesis that U is isometric implies that

(15.13.4.5)            (x\y- U-y) + (x- U-x\U-y) = Q

for all x, y in F; hence, if x — U • x = 0, x is orthogonal to G, and since G is
dense in E we have x = 0. Next, let us show that H defined by (15.13.4.3) is
hermitian. If x, y e G, then we may write x = u — U - u and y = v — U - v,
with w, v in F; hence, as (U - u \ U - v) = (u \ i?), we obtain

(H-x\y)= (i(u +U-u)\v-U-v)

which proves our assertion. Next we show that His closed. lf(yn) is a sequence
of points in G which tends to y e E and is such that the sequence (H • yn)
converges to z e E, then the sequence of points

which belong to F, converges to x = z + iy e F; hence, as 17is continuous on
F, the sequence of points (/ — U) • xn = 2iyn converges to a point in G. This
shows that y e G and that (/ — U) • x = 2iy. On the other hand, the sequence
of points (/ + U) - xn = 2H • yn converges to (/ + U) • x = 2z for the same
reason, and therefore H - y = z. Hence H is closed. Finally, if x = u - U • u
with u e F, then we have H - x = i(u + U • z/), and it is immediately verified
that K-t/= C7-U.

The isometric linear mapping V defined in (15.13.4.1) is called the Cayley
transform of H. Proposition (15.13.4) reduces the study of hermitian opera-
tors to that of their Cayley transforms.a positive self-adjoint operator jR such that dom(JR) = dom(T) and an isometry
```